# How do you Find exponential decay rate?

See below.

Exponential decays typically start with a differential equation of the form:

This is the general form of the exponential decay formula and will typically have graphs that look like this:

graph{e^-x [-1.465, 3.9, -0.902, 1.782]}

Perhaps an example might help?

Rearrange to get:

For the next part:

Now for the last part, the decay rate is already defined a way back at the very start, simply evaluate it at the given time:

The idea is to start with differential equation above, which gives the decay rate, and solve it to get the population at any given time.

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To find the exponential decay rate, you can use the formula:

[ A(t) = A_0 \times e^{-rt} ]

Where:

- ( A(t) ) is the amount at time ( t ).
- ( A_0 ) is the initial amount (at ( t = 0 )).
- ( r ) is the decay rate (which you want to find).
- ( t ) is the time.

To find the decay rate ( r ), you need to know:

- The initial amount ( A_0 ).
- The amount ( A(t) ) at a specific time ( t ).

The decay rate ( r ) can be found using the formula:

[ r = \frac{-\ln\left(\frac{A(t)}{A_0}\right)}{t} ]

Where:

- ( \ln ) denotes the natural logarithm.

To find ( r ):

- Substitute the given values of ( A(t) ) and ( A_0 ) into the formula.
- Calculate ( \ln\left(\frac{A(t)}{A_0}\right) ).
- Divide the result by ( t ) to find ( r ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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